Decision tree

• Split sample sucessively into smaller subsamples by answering “yes-no” questions.
• Each question is based on a single variable: is it above a threshold?
• Split a specified number of times (depth). Final subsamples are called leaves.
• The prediction for each observation is the mean of the training observations that end up in the same leaf.
• Variable to split on and threshold are chosen each time to minimize the SSE after the split.

Illustration

Random forest

• Multiple trees fit to random data
• Data for each tree is a bootstrapped sample:
  • random selection of rows (with replacement)
  • same size as original sample
• Prediction is average of predictions of the trees
• Hyperparameters = number of trees and depth of trees

Gradient boosting

• Multiple trees
• First tree fit to data
• Second tree fit to errors from first tree
• Third tree fit to errors from second tree, …
• Prediction is sum of predictions
• Hyperparameters = number of trees and depth of trees

Multi-layer perceptrons

• A multi-layer perceptron (MLP) consists of “neurons” arranged in layers.
• A neuron is a mathematical function. It takes inputs \(x_1, \ldots, x_n\), calculates a function \(y=f(x_1, \ldots, x_n)\) and passes \(y\) to the neurons in the next level.
• The inputs in the first layer are the predictors.
• The inputs in successive layers are the calculations from the prior level.
• The last layer is a single neuron that produces the output.

Illustration

• The output is a function of the 5 numbers calculated in the hidden layer.

Rectified linear units

• The usual function for the neurons (except in the last layer) is \[ y = \max(0,b+w_1x_1 + \cdots + w_nx_n)\] Parameters \(b\) (called bias) and \(w_1, \ldots w_n\) (called weights) are different for different neurons.
• This function is called a rectified linear unit (RLU). It’s like an option payoff.
• Last layer uses a linear function \[ y = b+w_1x_1 + \cdots + w_nx_n\]

Analogy to neurons firing

If \(w_i>0\) and \(b<0\), then \(y>0\) only when \(x_i\) are large enough.

A neuron fires when it is sufficiently stimulated by signals from other neurons (in prior layer).

Deep learning

• Deep learning means a neural network with many layers.
• Deep learning is behind facial recognition, self-driving cars, …
• Need specialized library, probably TensorFlow (from Google) or PyTorch (from Facebook)
• And probably need a graphical processing unit (GPU) – i.e., run on a video card
• Can often start from a pretrained model